What weapons (and armor) do I care about?

D&D’s gear fetish attempts to model (psychologically and statistically) the differences in how your hero slays her foes.

5e gets it wrong in my opinion. The bones are fine; it’s the detail that needs work.
For one thing, in the finished rules, damage type just doesn’t matter much, so disambiguating on damage type doesn’t do a lot.
For another, cost and weight are also relatively fungible; beyond first level a few gp or even tens if gp don’t matter, and lighter isn’t hugely important since encumbrance is rare and the weights are so similar.

For a final thing, in the quest to strike a balance between simplicity and differentiation, they left a lot of arbitrage. Tridents are spears that only fighters can use. Daggers are amazing, even in the hands of a wizard. Scimitars and short swords are the same thing.
For some reason, quarterstaves can be used one handed.

The simplest fix which I can think of is:
1) rename the d6 scimitar to the kukri.
2) reflavor the d8 rapier to the sidesword; elves have proficiency in it instead of longsword; it is a light weapon when paired with a d4 light weapon. It is also the scimitar.
3) give the flail the versatile property.
4) give the trident a d8 base, versatile d10. What, and thrown? Sure. It’s a big weapon, and the magic ones tend to not be Returning. If this marginal utility really bothers you, you might make it Heavy too.
5) rename the club “wand”. A heavy caveman club with a rock at the end is a mace or hammer.
6) give the mace the versatile property. It is also the “rod”, the weapon the devs were thinking of when they made staff versatile, about chest-high.
7) replace the staff with a d6 two handed reach weapon.
8) the spiked-chain is pretty much just a polearm; add it in.

Now, even this isn’t ideal; class-based damage would have done for that. Also, my ideal rev gives weapons increasing properties with better proficiencies; for instance daggers being light might be a property of martial proficiency.
But this is pretty good.